Question 6 of 6: Acceptance Sampling, MIL-STD-105E, and a Single Sampling Plan

Question 6: Acceptance Sampling, MIL-STD-105E, and a Single Sampling Plan (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) AQL-based sampling, producer's/consumer's risk, and the main features of MIL-STD-105E

MIL-STD-105E is AQL-based: every plan in it is indexed by an Acceptable Quality Level, the poorest process average (fraction nonconforming) that the SCHEME as a whole is still willing to treat as satisfactory. This does not mean any single lot at exactly the AQL is guaranteed acceptance — it means the plan's operating characteristic (OC) curve is chosen so lots from a process that genuinely runs AT OR BETTER than the AQL are accepted with high probability, and the scheme's switching rules (normal → tightened after repeated rejections, and back, plus optional reduced inspection) give an ongoing, lot-after-lot incentive for a supplier to keep running at or below the AQL, rather than protecting the consumer against any single bad lot in isolation.

Both risks are read directly off the OC curve of the chosen plan $(n,Ac)$: the producer's risk $\alpha=1-P_a(AQL)$ is the probability that a genuinely GOOD lot (fraction nonconforming $=AQL$) is nevertheless rejected by chance sampling variation — a cost borne by the producer, since acceptable product is refused. The consumer's risk $\beta=P_a(LQL)$ is the probability that a genuinely BAD lot (fraction nonconforming at some stated unacceptable Limiting Quality Level) is nevertheless accepted — a cost borne by the consumer. Both are computed from the acceptance probability function $P_a(p)=P(X\le Ac)$ for the plan's sample size $n$ and acceptance number $Ac$, using the binomial distribution exactly or, as is standard practice for small $p$ and moderate $n$, the Poisson approximation with $\lambda=np$.

The main features of MIL-STD-105E: lot size and a chosen inspection level (general levels I/II/III, trading off discriminating power against sample size, plus special levels S-1–S-4 for expensive or destructive tests) together select a sample-size CODE LETTER from a standard table; AQL-indexed master tables (separate ones for normal, tightened, and reduced inspection) then give the sample size $n$ and acceptance/rejection numbers $Ac/Re$ for that code letter and AQL; and a set of SWITCHING RULES moves the scheme between normal, tightened (triggered by a run of rejected lots, giving tougher acceptance numbers), and reduced inspection (triggered by a sustained run of accepted lots, giving a smaller sample for the same AQL) — a self-adjusting mechanism that protects the consumer from a supplier whose quality has degraded while rewarding a supplier with a demonstrated track record. The standard supports single, double, and multiple sampling plans built on the same AQL/code-letter framework.

(b) Traditional vs. sequential sampling; when to prefer Dodge-Romig; AOQL vs. LTPD

Traditional (fixed-$n$) acceptance sampling draws one sample of predetermined size and decides accept/reject immediately. Its advantages are administrative simplicity, a KNOWN and bounded sample size and inspection time/cost per lot, and ease of auditing/explaining to both parties. Sequential sampling (e.g. Wald's sequential probability ratio test) inspects units one at a time (or in small increments), comparing the cumulative result after each against accept/reject/continue boundaries; its advantage is a SMALLER average total sample size than a fixed-$n$ plan for the SAME $(\alpha,\beta)$ protection, since it can stop early whenever the evidence is already decisive. Its disadvantages are greater administrative complexity (continuous testing rather than a single fixed inspection event), an unpredictable total sample size and inspection duration/cost from lot to lot, and unsuitability whenever testing is destructive or slow (each additional unit tested is costly, and an open-ended sequential test can drag out disproportionately). Traditional fixed-$n$ plans are therefore preferred whenever inspection must fit a fixed schedule, when destructive testing makes an unpredictable sample size expensive, or simply for administrative simplicity.

Dodge-Romig plans are indexed by LTPD (or by AOQL) and are explicitly DESIGNED to minimize the average total inspection (ATI) for a GIVEN, known process average, protecting the CONSUMER (a fixed $\beta$ at the stated LTPD) rather than sustaining an ongoing producer incentive relationship the way MIL-105E's AQL/switching-rule scheme does. Dodge-Romig is the better choice when the primary concern is bounding the WORST-case outgoing quality on a single critical characteristic (a consumer-protection focus) and the incoming process average is reasonably well known/stable, so that the ATI-minimization the plans are built around is actually realized; MIL-105E's AQL scheme is preferable when the goal is an ongoing, evolving relationship with a supplier across MANY lots, where the switching rules' incentive effect matters more than minimizing inspection on any one lot.

AOQL (average outgoing quality limit) is the WORST long-run AVERAGE outgoing fraction nonconforming that can result from a rectifying-inspection sampling scheme (where every rejected lot is 100% inspected and its nonconforming units replaced or removed, and any nonconforming units found in an accepted lot's sample are likewise corrected), MAXIMIZED over every possible incoming process quality — it is a property of the whole rectification scheme, a long-run ceiling on outgoing defect rate regardless of how bad incoming lots might be. LTPD (or LQL) is a single POINT on the OC curve — the incoming lot quality at which the consumer wants the acceptance probability held down to a small stated $\beta$ — a statement about the risk of accepting ONE bad lot, not a long-run average. The two therefore answer different questions (a long-run AVERAGE outgoing-quality guarantee vs. a single-lot ACCEPTANCE-probability guarantee), and the same $(n,Ac)$ plan implies both an AOQL and an LTPD/$\beta$ pair, but a plan can be DESIGNED (chosen) against either criterion, with generally different resulting sample sizes.

(c) MIL-STD-105E single sampling plan and its producer's/consumer's risk

Given. Normal inspection, general inspection level II, lot size $N=1{,}500$, $AQL=0.40\%$ nonconforming, $LQL=3\%$.

Find. The single sampling plan ($n$, $Ac$, $Re$), and the producer's risk $\alpha$ (at $p=AQL$) and consumer's risk $\beta$ (at $p=LQL$).

Approach. Look up the sample-size code letter from the lot-size/inspection-level table, read $(n,Ac,Re)$ off the normal-inspection master table (Table II-A) at that code letter and the stated AQL, then compute $\alpha$ and $\beta$ from the Poisson approximation to the plan's OC curve.

1. Sample-size code letter. Lot size 1,500 falls in the range 1,201–3,200; at general inspection level II this gives code letter $\boxed{K}$ (sample size code-letter table).
2. Sample size, $Ac$, $Re$. Code letter K corresponds to sample size $n=125$; reading the AQL=0.40% column of the normal-inspection master table (Table II-A) at code K gives $$n=\boxed{125}\qquad Ac=\boxed{1}\qquad Re=\boxed{2}$$ i.e. inspect 125 of the 1,500 items; accept the lot if 1 or fewer nonconforming units are found, reject if 2 or more are found.
3. Producer's risk at $p=AQL=0.40\%$. Using the Poisson approximation with $\lambda=np$: $$\lambda_{AQL}=125(0.0040)=0.500$$ $$P_a(AQL)=P(X\le1\mid\lambda=0.500)=e^{-0.5}(1+0.5)=0.6065(1.5)=0.9098$$ $$\alpha=1-P_a(AQL)=1-0.9098=\boxed{0.090}\ (9.0\%)$$
4. Consumer's risk at $p=LQL=3\%$. $$\lambda_{LQL}=125(0.03)=3.75$$ $$\beta=P_a(LQL)=P(X\le1\mid\lambda=3.75)=e^{-3.75}(1+3.75)=0.02352(4.75)=\boxed{0.112}\ (11.2\%)$$